## Problems

Chapter 12 Problems 2 and 3 (1st Edition)

Chapter 9 Problems 5 and 6 (2nd Edition)

## Solutions

### 12.2 (9.5 in 2nd Edition)

This question, it seems, assumes that the base salary (s) cannot be negative.
Then, to induce high effort (incentive compatibility):
.2b ≥ $50,000 → b ≥ $250,000

Next, check the participation constraint:
.8b+s = .8($250,000)+s = $200,000+s ≥ $150,000

this is satisfied for all s >= -$50,000 so set s=0.
Now, what is the firm's expected profit?
.8($400,000-$250,000) = .8($150,000) = $120,000

What if we didn't induce high effort? To get only routine effort,
we need to pay only $100,000 and no bonus. The firm's expected profit would be:
.6($400,000) - $100,000 = $240,000 - $100,000 = $140,000

Hence, the cost of the information asymmetry is too high,
and the firm is better off not trying to induce high effort.
### 12.3 (9.6 in 2nd Edition)

(a) To achieve separation, we need qualified workers to prefer the Good job
at a salary of 100 to a Bad job at a salary of 10. Similarly, unqualified workers
should prefer the bad job to the good job:
Qualified: | 100 - n^{2}/2 > 10 | → | n^{2}/2 < 90
| → | n^{2} < 180 |

Unqualified: | 100 - n^{2} < 10 | → | n^{2} > 90 |

Hence, the smallest n^{2} is 90. Since n has to be an integer, n=10 is the minimum value.
(b) Since the company has to pay for each type of job a wage equal to expected production:

Production in good jobs: | 0.6(100) + 0.4(0) = | 60 |

Production in bad jobs: | 10 |

Hence, since the salary is higher in good jobs, both types will accept the good job,
and no worker will agree to take the bad job. The unqualified workers are definitely
better off (earning 60 instead of 10). Strangely enough, the qualified workers are
better off, too. In the separating equilibrium, qualified workers earned 100,
but the signal cost them:
n^{2}/2 = (10)^{2}/2 = 100/2 = 50

So their surplus was only 100-50=50.
Now, they do not invest resources in signaling, but earn 60.